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Structured rhombic triacontahedral numbers (vertex structure 11).
3

%I #21 Sep 20 2022 02:37:29

%S 1,32,143,384,805,1456,2387,3648,5289,7360,9911,12992,16653,20944,

%T 25915,31616,38097,45408,53599,62720,72821,83952,96163,109504,124025,

%U 139776,156807,175168,194909,216080,238731,262912,288673,316064,345135,375936,408517,442928

%N Structured rhombic triacontahedral numbers (vertex structure 11).

%C Also structured triakis icosahedral numbers (vertex structure 11) (cf. A100172 = alternate vertex).

%H Vincenzo Librandi, <a href="/A100164/b100164.txt">Table of n, a(n) for n = 1..5000</a>

%F a(n) = (1/6)*(50*n^3 - 60*n^2 + 16*n) = (1/3)*n*(5*n-2)*(5*n-4).

%F From _Jaume Oliver Lafont_, Sep 08 2009: (Start)

%F a(n) = (5*(n-1) + 1)*(5*(n-1) + 3)*(5*(n-1) + 5)/15.

%F G.f.: x*(1 + 28*x + 21*x^2)/(1-x)^4. (End)

%F Sum_{n>=1} 1/a(n) = 3*sqrt((25-2*sqrt(5))/5)*Pi/16 + 9*sqrt(5)*log(phi)/16 - 15*log(5)/32, where phi is the golden ratio (A001622). - _Amiram Eldar_, Sep 20 2022

%t a[n_] := (n*(5*n - 2)*(5*n - 4))/3; Array[a, 30] (* _Amiram Eldar_, Sep 20 2022 *)

%o (Magma) [(1/6)*(50*n^3-60*n^2+16*n): n in [1..40]]; // _Vincenzo Librandi_, Jul 25 2011

%Y Cf. A100165 (alternate vertex), A100145 for more on structured polyhedral numbers.

%Y Cf. A001622, A100172.

%K easy,nonn

%O 1,2

%A James A. Record (james.record(AT)gmail.com), Nov 07 2004